Tracking of triangular or sawtooth waveforms is a major difficulty for achieving high-speed operation in many scanning applications such as scanning probe microscopy. Such non-smooth waveforms contain high order harmonics of the scan frequency that can excite mechanical resonant modes of the positioning system, limiting the scan range and bandwidth. Hence, fast raster scanning often leads to image distortion. This paper proposes analysis and design methodologies for a nonlinear and smooth closed curve, known as Lissajous pattern, which allows much faster operations compared to the ordinary scan patterns. A simple closed-form measure is formulated for the image resolution of the Lissajous pattern. This enables us to systematically determine the scan parameters. Using internal model controllers (IMC), this non-raster scan method is implemented on a commercial atomic force microscope driven by a low resonance frequency positioning stage. To reduce the tracking errors due to actuator nonlinearities, higher order harmonic oscillators are included in the IMC controllers. This results in significant improvement compared to the traditional IMC method. It is shown that the proposed IMC controller achieves much better tracking performances compared to integral controllers when the noise rejection performances is a concern.

For a constrained flexible beam actuated at one end, we introduce three outputs to achieve fast regulation of the nonminimum phase endpoint force. To this end, three auxiliary signals, i.e., angular displacement, moment, and shear force at the hub are utilized to define new outputs, which are referred to as redefined outputs. Each redefined output is composed of the endpoint force augmented by the time derivative of an auxiliary signal measured at the hub. It is shown that using the redefined outputs along with proper controllers can significantly improve control performance, such as bandwidth, robustness, and error performance. Analytical results are presented that illustrate how bandwidth improvements in a force control loop can be achieved. The results are further verified through simulations and experimental studies.

We address design, implementation, and characterization of semiconductor strain gauges as stage displacement sensors with small footprint for piezo-driven nanopositioners in differential actuation mode. The strain gauges are not collocated with the stage of the nanopositioner. They are attached to piezo stack actuators in both longitudinal and transverse directions. We address two differential configurations for the displacement sensing. The first configuration uses all strain gauges and provides three output signals. Two of them are proportional to the length variations of the piezo actuators, which are considered as conventional non-differential sensing method. The last output in the first configuration is the first proposed differential sensor. In the second configuration, we propose an alternative differential sensor using only the longitudinal strain gauges and a simpler readout circuit. The results indicate that in the differential actuation mode, a differential sensing strategy provides much better accuracy than the conventional non-differential schemes. Using a laser interferometer as an independent collocated sensor for stage displacement, we obtained constant calibration factors both proposed differential sensors and characterized their sensing accuracies.

In this paper, we propose a novel approach for tracking of piecewise linear trajectories, such as triangular and staircase waveforms. We derive state and input transformations, which result in closed-loop error dynamics driven by a series of impulses. The proposed control structure takes the form of an output-feedback-feedforward system that is straightforward to implement. In contrast to the recently proposed tracking control methods for such trajectories, the closed-loop stability is not affected by the frequency of the desired triangular reference. The method is implemented on a nanopositioner serving as the scanning stage of an atomic force microscope.